Question

Find an expression for the function whose graph is the line segment joining the points (1, -1) and (5, 5). (Assume that these points are in the form (x, f(x)).)

Answer 1

Namely, what's the equation of a line that passes through (1,-1) and (5,5)

well, first off let's find its slope, and the plug all those values in the point-slope form.

$\bf \begin{array}{lllll} &x_1&y_1&x_2&y_2\\ % (a,b) &({{ 1}}\quad ,&{{ -1}})\quad % (c,d) &({{ 5}}\quad ,&{{ 5}}) \end{array} \\\\\\ % slope = m slope = {{ m}}= \cfrac{rise}{run} \implies \cfrac{{{ y_2}}-{{ y_1}}}{{{ x_2}}-{{ x_1}}}\implies \cfrac{5-(-1)}{5-1}\implies \cfrac{5+1}{5-1} \\\\\\ \cfrac{6}{4}\implies \cfrac{3}{2}$

$\bf \stackrel{\textit{point-slope form}}{y-{{ y_1}}={{ m}}(x-{{ x_1}})}\implies y-(-1)=\cfrac{3}{2}(x-1) \\\\\\ y+1=\cfrac{3}{2}x-\cfrac{3}{2}\implies y=\cfrac{3}{2}x-\cfrac{3}{2}-1\implies y=\cfrac{3}{2}x-\cfrac{5}{2}$